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Analysis of direct discontinuous Galerkin methods for multi-dimensional convection–diffusion equations

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Abstract

We provide a framework for the analysis of the direct discontinuous Galerkin (DDG) methods for multi-dimensional convection–diffusion equations subject to various boundary conditions. A key tool is the global projection constructed by the DDG scheme applied to an associated elliptic problem. Such projection is well-defined for a class of diffusive flux parameters, and the optimal projection error in \(L^2\) is obtained with an arbitrary locally regular partition of the domain and for an arbitrary degree of polynomials. This results in the optimal \(L^2\) error for the DDG method to the elliptic problem, and further leading to the optimal \(L^2\) error for the DDG method to the convection–diffusion problem.

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References

  1. Arnold, D.N.: An interior penalty finite element method with discontinuous elements. SIAM J. Numer. Anal. 19(4), 742–760 (1982)

    Article  MathSciNet  MATH  Google Scholar 

  2. Arnold, D.N., Brezzi, F., Cockburn, B., Marini, L.D.: Unified analysis of discontinuous Galerkin methods for elliptic problems. SIAM J. Numer. Anal. 39(5), 1749–1779 (electronic) (2001/02)

  3. Baker, G.A.: Finite element methods for elliptic equations using nonconforming elements. Math. Comput. 31, 45–59 (1977)

    Article  MathSciNet  MATH  Google Scholar 

  4. Bassi, F., Rebay, S.: A high-order accurate discontinuous finite element method for the numerical solution of the compressible Navier–Stokes equations. J. Comput. Phys. 131(2), 267–279 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  5. Baumann, C.E., Oden, J.T.: A discontinuous \(hp\) finite element method for convection–diffusion problems. Comput. Methods Appl. Mech. Eng. 175(3–4), 311–341 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  6. Castillo, P.: An optimal error estimate for the Local Discontinuous Galerkin method. In: Cockburn, B., Karniadakis, G., Shu, C.-W. (eds.) Discontinuous Galerkin Methods: Theory, Computation and Applications. Lecture Notes in Computational Science and Engineering, vol. 11, pp. 285–290. Springer, Berlin (2000)

    Chapter  Google Scholar 

  7. Castillo, P.: Performance of discontinuous Galerkin methods for elliptic PDEs. SIAM J. Sci. Comput. 24, 524–547 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  8. Celiker, F., Cockburn, B.: Superconvergence of the numerical traces of discontinuous Galerkin and hybridized methods for convection–diffusion problems in one space dimension. Math. Comput. 76(257), 67–96 (2007). (electronic)

    Article  MathSciNet  MATH  Google Scholar 

  9. Castillo, P., Cockburn, B., Perugia, I., Schötzau, D.: An a priori error analysis of the local discontinuous Galerkin method for elliptic problems. SIAM J. Numer. Anal. 38(5), 1676–1706 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  10. Castillo, P., Cockburn, B., Schötzau, D., Schwab, C.: Optimal a priori error error estimates for the hp-version of the LDG method for convection–diffusion problems. Math. Comput. 71, 455–478 (2002)

    Article  MATH  Google Scholar 

  11. Cockburn, B., Dong, B., Guzman, J.: Optimal convergence of the original DG method for the transport-reaction equation on special meshes. SIAM J. Numer. Anal. 46, 1250–1265 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  12. Ciarlet, Philippe G.: The Finite Element Method for Elliptic Problems. Studies in Mathematics and its Applications, vol. 4. North-Holland Publishing Co., Amsterdam (1978)

    Google Scholar 

  13. Cockburn, B., Dawson, C.: Approximation of the velocity by coupling discontinuous Galerkin and mixed finite element methods for flow problems. Comput. Geosci. 6(3–4), 505–522 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  14. Cockburn, B., Hou, S., Shu, C.-W.: The Runge–Kutta local projection discontinuous Galerkin finite element method for conservation laws. IV. The multidimensional case. Math. Comput. 54(190), 545–581 (1990)

    MathSciNet  MATH  Google Scholar 

  15. Cockburn, B., Kanschat, G., Schotzau, D.: A locally conservative LDG method for the incompressible Navier–Stokes equations. Math. Comput. 74(251), 1067–1095 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  16. Cockburn, B., Lin, S.Y., Shu, C.-W.: TVB Runge–Kutta local projection discontinuous Galerkin finite element method for conservation laws. III. One-dimensional systems. J. Comput. Phys. 84(1), 90–113 (1989)

    Article  MathSciNet  MATH  Google Scholar 

  17. Cheng, J., Liu, T.G., Luo, H.: A hybrid reconstructed discontinuous Galerkin method for compressible flows on arbitrary grids. Comput. Fluids 139, 68–79 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  18. Cao, W.-X., Liu, H., Zhang, Z.-M.: Superconvergence of the direct discontinuous Galerkin method for convection–diffusion equations. Numer. Methods. PDEs 33(1), 290–317 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  19. Cheng, Y., Meng, X., Zhang, Q.: Application of generalized Gauss–Radau projections for the local discontinuous Galerkin method for linear convection–diffusion equations. Math. Comput. 86, 1233–1267 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  20. Cockburn, B., Shu, C.-W.: TVB Runge–Kutta local projection discontinuous Galerkin finite element method for conservation laws. II. General framework. Math. Comput. 52(186), 411–435 (1989)

    MathSciNet  MATH  Google Scholar 

  21. Cockburn, B., Shu, C.-W.: The local discontinuous Galerkin method for time-dependent convection–diffusion systems. SIAM J. Numer. Anal. 35(6), 2440–2463 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  22. Cheng, Y.-D., Shu, C.-W.: A discontinuous Galerkin finite element method for time dependent partial differential equations with higher order derivatives. Math. Comput. 77(262), 699–730 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  23. Cockburn, B., Shu, C.-W.: The Runge–Kutta discontinuous Galerkin method for conservation laws. V. Multidimensional systems. J. Comput. Phys. 141(2), 199–224 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  24. Cheng, Jian, Yang, Xiaoquan, Liu, Xiaodong, Liu, Tiegang, Luo, Hong: A direct discontinuous Galerkin method for the compressible Navier–Stokes equations on arbitrary grids. J. Comput. Phys. 327, 484–502 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  25. Douglas Jr. J., Dupont, T.: Interior penalty procedures for elliptic and parabolic Galerkin methods. In: Computing Methods in Applied Sciences (Second International Symposium, Versailles, 1975), Lecture Notes in Physics, vol. 58, pp. 207–216. Springer, Berlin (1976)

  26. Gassner, G., Lörcher, F., Munz, C.D.: A contribution to the construction of diffusion fluxes for finite volume and discontinuous Galerkin schemes. J. Comput. Phys. 224(2), 1049–1063 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  27. Huang, Y.-Q., Liu, H., Yi, N.-Y.: Recovery of normal derivatives from the piecewise \(L^2\) projection. J. Comput. Phys. 231, 1230–1243 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  28. Hesthaven, J.S., Warburton, T.: Nodal Discontinuous Galerkin Methods: Algorithms, Analysis, and Applications. Springer, New York (2007)

    MATH  Google Scholar 

  29. Johnson, C., Pitakäranta, J.: An analysis of the discontinuous Galerkin method for a scalar hyperbolic problem. Math. Comput. 46, 1–26 (1986)

    Article  Google Scholar 

  30. Lu, W.-Y., Huang, Y.-Q., Liu, H.: Mass preserving direct discontinuous Galerkin methods for Schrödinger equations. J. Comput. Phys. 282(1), 210–226 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  31. van Leer, B., Nomura, S.: scontinuous Galerkin for diffusion. In: Proceedings of 17th AIAA Computational Fluid Dynamics Conference (June 6 2005), AIAA-2005-5108

  32. LeSaint, P., Raviart, P.A.: On a finite element method for solving the neutron transport equation. In: de Boor, C. (ed.) Mathematical Aspects of Finite Elements in Partial Differential Equations, pp. 89–145. Academic Press, New York (1974)

    Chapter  Google Scholar 

  33. Liu, H.: Optimal error estimates of the direct discontinuous Galerkin method for convection–diffusion equations. Math. Comput. 84, 2263–2295 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  34. Liu, H., Wang, Z.-M.: An entropy satisfying discontinuous Galerkin method for nonlinear Fokker–Planck equations. J. Sci. Comput. 68, 1217–1240 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  35. Liu, H., Wang, Z.-M.: A free energy satisfying discontinuous Galerkin method for Poisson–Nernst–Planck systems. J. Comput. Phys. 238, 413–437 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  36. Liu, H., Wen, H.-R.: Error estimates for the AEDG method to one-dimensional linear convection–diffusion equations. Math. Comput. 87, 123–148 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  37. Liu, H., Yan, J.: The Direct Discontinuous Galerkin (DDG) methods for diffusion problems. SIAM J. Numer. Anal. 47(1), 675–698 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  38. Liu, H., Yan, J.: The Direct Discontinuous Galerkin (DDG) method for diffusion with interface corrections. Commun. Comput. Phys. 8(3), 541–564 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  39. Liu, H., Yu, H.: Maximum-principle-satisfying third order discontinuous Galerkin schemes for Fokker-Planck equations. SIAM J. Sci. Comput. 36(5), A2296–A2325 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  40. Liu, H., Yu, H.: The entropy satisfying discontinuous Galerkin method for Fokker–Planck equations. J. Sci. Comput. 62, 803–830 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  41. Oden, J.T., Babuška, I., Baumann, C.E.: A discontinuous \(hp\) finite element method for diffusion problems. J. Comput. Phys. 146(2), 491–519 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  42. Osher, S.: Riemann solvers, the entropy condition, and difference approximations. SIAM J. Numer. Anal. 21(2), 217–235 (1984)

    Article  MathSciNet  MATH  Google Scholar 

  43. Peterson, T.: A note on the convergence of the discontinuous Galerkin method for a scalar hyperbolic equation. SIAM J. Numer. Anal. 28(1), 133–140 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  44. Peraire, J., Persson, P.-O.: The compact discontinuous Galerkin (CDG) method for elliptic problems. SIAM J. Sci. Comput. 30(4), 1806–1824 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  45. Reed, W.H., Hill, T.R.: Triangular mesh methods for the neutron transport equation. Technical Report LA-UR-73-479, Los Alamos Scientific Laboratory, 1973

  46. Richter, G.R.: An optimal-order error estimate for the discontinuous Galerkin method. Math. Comput. 50, 75–88 (1988)

    Article  MathSciNet  MATH  Google Scholar 

  47. Rivière, B.: Discontinuous Galerkin Methods for Solving Elliptic and Parabolic Equations. SIAM, Philadelphia (2008)

    Book  MATH  Google Scholar 

  48. Riviere, B., Wheeler, M.F.: A discontinuous Galerkin method applied to nonlinear parabolic equations. In: Cockburn, B., Karniadakis, G., Shu, C.-W. (eds.) Discontinuous Galerkin Methods: Theory, Computation and Applications. Lecture Notes in Computational Science and Engineering, vol. 11, pp. 231–244. Springer, Berlin (2000)

    Chapter  Google Scholar 

  49. Shu, C.-W.: Discontinuous Galerkin methods: general approach and stability. In: Numerical Solutions of Partial Differential Equations, Advanced Courses Mathematics. CRM Barcelona, pp. 149–201. Birkhäuser, Basel (2009)

  50. Wheeler, M.F.: An elliptic collocation-finite element method with interior penalties. SIAM J. Numer. Anal. 15, 152–161 (1978)

    Article  MathSciNet  MATH  Google Scholar 

  51. Wang, H.J., Shu, C.-W., Zhang, Q.: Stability and error estimates of local discontinuous Galerkin methods with implicit-explicit time-marching for advection-diffusion problems. SIAM J. Numer. Anal. 53(1), 206–227 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  52. Wang, H.J., Zhang, Q.: Error estimate on a fully discrete local discontinuous Galerkin method for linear convection–diffusion problems. J. Comput. Math. 31(3), 283–307 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  53. Xu, Y., Shu, C.-W.: Error estimates of the semi-discrete local discontinuous Galerkin method for nonlinear convection–diffusion and KdV equations. Comput. Methods Appl. Mech. Eng. 196, 3805–3822 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  54. Yin, P.-M., Huang, Y.-Q., Liu, H.: A direct discontinuous Galerkin (DDG) method for solving the nonlinear Poisson–Boltzmann equation. Commun. Comput. Phys. 16(2), 491–515 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  55. Yu, H., Liu, H.: Third order maximum-principle-satisfying DG schemes for convection–diffusion problems with anisotropic diffusivity. J. Comput. Phys. 391, 14–36 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  56. Zhang, F., Cheng, J., Liu, T.G.: A direct discontinuous Galerkin method for the incompressible Navier–Stokes equations on arbitrary grids. J. Comput. Phys. 362, 95–113 (2019)

    MathSciNet  MATH  Google Scholar 

  57. Zhang, Q., Shu, C.-W.: Error estimates to smooth solutions of Runge–Kutta discontinuous Galerkin methods for scalar conservation laws. SIAM J. Numer. Anal. 42, 641–666 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  58. Zhang, Q., Shu, C.-W.: Error estimates to smooth solutions of Runge–Kutta discontinuous Galerkin methods for symmetrizable systems of conservation laws. SIAM J. Numer. Anal. 44, 1703–1720 (2006)

    Article  MathSciNet  MATH  Google Scholar 

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Acknowledgements

This research was partially supported by the National Science Foundation under Grants DMS1312636 and DMS1812666.

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  1. Department of Mathematics, Iowa State University, Ames, IA, 50010, USA

    Hailiang Liu

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Liu, H. Analysis of direct discontinuous Galerkin methods for multi-dimensional convection–diffusion equations. Numer. Math. 147, 839–867 (2021). https://doi.org/10.1007/s00211-021-01183-x

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